Looking at data: distributions - Displaying distributions with graphs IPS section 1.1 © 2006 W.H. Freeman and Company (authored by Brigitte Baldi, University of California-Irvine; adapted by Jim Brumbaugh-Smith, Manchester College)

Objectives Displaying distributions with graphs  Recognize numerical vs. categorical data  Construct graphs representing a distribution of numerical data  Histograms  Stemplots  Describe overall patterns in numerical data  Identify exceptions to overall patterns  Discuss pros and cons of histograms vs. stemplots

Terminology  Individual (or “observation”)  Variable  numerical (or “quantitative”)  categorical (or “qualitative)  Value  Frequency  absolute  relative  Frequency table  Distribution

Terminology (cont’d)  Graphs for quantitative data  Histogram  Stemplot (or “stem-and-leaf diagram”)  Boxplot (section 1.2)  Symmetric  Skewed right (or “positively skewed”)  Skewed left (or “negatively skewed”)  Peak (or “mode”)  Unimodal vs. Bimodal  Outlier

Variables In a study, we collect data from individuals, more formally known as observations. Observations can be people, animals, plants, or any object or process of interest. A variable is a characteristic that varies among individuals in a population or in a sample (a subset of a population). Example: age, height, blood pressure, ethnicity, leaf length, first language The distribution of a variable tells us what values the variable takes and how often it takes on these values. A distribution can also be thought of as the pattern of variation seen in the data.

Two types of variables  Variables can be either numerical (a/k/a quantitative) …  Something that can be counted or measured for each individual and then added, subtracted, averaged, etc. across individuals in the population.  Example: How tall you are, your age, your blood cholesterol level, the number of credit cards you own  … or categorical (a/k/a qualitative).  Something that falls into one of several categories. What can be computed is the count or proportion of individuals in each category.  Example: Your blood type (A, B, AB, O), your hair color, your ethnicity, whether you paid income tax last tax year or not

How do you know if a variable is categorical or quantitative? Ask:  What are the individuals in the sample?  What is being recorded about those individuals?  Is that a number (“quantitative”) or a statement (“categorical”)? Individuals in sample DIAGNOSISAGE AT DEATH Patient AHeart disease56 Patient BStroke70 Patient CStroke75 Patient DLung cancer60 Patient EHeart disease80 Patient FAccident73 Patient GDiabetes69 Quantitative Each individual is attributed a numerical value. Categorical Each individual is assigned to one of several categories.

Ways to chart categorical data Because the variable is categorical, the data in the graph can be ordered any way we want (alphabetical, by increasing value, by year, by personal preference, etc.)  Bar graphs Each category is represented by a bar.  Pie charts Peculiarity: The slices must represent the parts of one whole.

Example: Top 10 causes of death in the United States 2001 RankCauses of deathCounts % of top 10s % of total deaths 1Heart disease700,14237%29% 2Cancer553,76829%23% 3Cerebrovascular163,5389%7% 4Chronic respiratory123,0136%5% 5Accidents101,5375%4% 6Diabetes mellitus71,3724%3% 7Flu and pneumonia62,0343% 8Alzheimer’s disease53,8523%2% 9Kidney disorders39,4802% 10Septicemia32,2382%1% All other causes629,96726% For each individual who died in the United States in 2001, we record what was the cause of death. The table above is a summary of that information.

Top 10 causes of deaths in the United States 2001 Bar graphs Each category is represented by one bar. The bar’s height shows the count (or sometimes the percentage) for that particular category. The number of individuals who died of an accident in 2001 is approximately 100,000.

Bar graph sorted by rank  Easy to analyze Top 10 causes of deaths in the United States 2001 Sorted alphabetically  Much less useful

Percent of people dying from top 10 causes of death in the United States in 2000 Pie charts Each slice represents a piece of one whole. The size of a slice depends on what percent of the whole this category represents.

Percent of deaths from top 10 causes Percent of deaths from all causes Make sure your labels match the data. Make sure all percents add up to 100.

Child poverty before and after government intervention—UNICEF, 1996 What does this chart tell you? The United States has the highest rate of child poverty among developed nations (22% of under 18). Its government does the least—through taxes and subsidies—to remedy the problem (size of orange bars and percent difference between orange/blue bars). Could you transform this bar graph to fit in 1 pie chart? In two pie charts? Why? The poverty line is defined as 50% of national median income.

Histograms  Vertical bar chart where horizontal axis is a numerical scale corresponding to the data.  Vertical axis represents frequency (how many) or relative frequency (what proportion)  “Peaks” correspond to commonly occurring data values.  “Valleys” and “tails” correspond to values which do not occur as frequently.  Most states have between 0 and 10 percent Hispanic residents  A very small number have between 25 and 45 percent.

Histograms The range of values that a variable can take is divided into equal size intervals. The histogram shows the number (i.e., frequency) of individual data points that fall in each interval. The first column represents all states with a percent Hispanic in their population between 0% and 4.99%. The height of the column shows how many states (27) have the percent Hispanic residents in this range. The last column represents all states with a percent Hispanic between 40% and 44.99%. There is only one such state: New Mexico, at 42.1% Hispanics.

Creating a histogram What “class size” should you use?  Use an appropriate number of classes − usually between 5 and 15 work well, depending on number of data values being represented.  Either too few or too many classes will obscure the pattern in the data.  Not so detailed that it is no longer summary  Avoid using many classes having frequency of only 0 or 1  Not overly summarized so that you lose all the information  rule of thumb: start with 5 to 10 classes Look at the distribution and refine your classes. (There isn’t a unique or “perfect” histogram.)

Guidelines for histograms  Label the horizontal axis with a consistent numerical scale. (Don’t leave any gaps in the scale or compress the scale toward the left or right side of the graph)  Use vertical bars of equal width.  Label the horizontal scale on the class boundaries.  An exception is when each bar corresponds to a single whole number. Then it is reasonable to label each bar at its midpoint.

Not summarized enough Too summarized Same data set

Interpreting histograms When describing the distribution of a quantitative variable, we look for the overall pattern and for striking deviations from that pattern. We can describe the overall pattern of a histogram by its shape, center, and spread. Histogram with a line connecting each column  too detailed Histogram with a smoothed curve highlighting the overall pattern of the distribution

Most common distribution shapes  A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. Symmetric distribution Complex, multimodal distribution  Not all distributions have a simple overall shape, especially when there are few observations. Skewed right distribution  A distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the extends much farther out than the right.

IMPORTANT NOTE: Your data are the way they are. Do not try to force them into a particular shape. It is a common misconception that if you have a large enough data set, the data will eventually turn out nice and symmetrical. Histogram of Drydays in 1995

AlaskaFlorida Outliers An important kind of deviation is an outlier. Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them. The overall pattern is fairly symmetrical except for two states clearly not belonging to the main trend. Alaska and Florida have unusual representation of the elderly in their population. A large gap in the distribution is typically a sign of an outlier.

More on Outliers  If outlier is incorrect data (e.g., data entry error, false response)  Correct it if possible.  Discard if absolutely sure it is wrong.  Discarding data might introduce a bias if uncorrectable errors tend to be mainly high (or mainly low) values.  If outlier is correct data consider effect on analysis  Which statistical technique is most appropriate?  How are conclusions affected by the outlier?  Some Other Strategies:  Refine population definition if unusual responses are not part of intended study group.  If sample size is quite small collect more data to see if any gaps fill in.

Stemplots (or “stem-and-leaf” diagrams) How to make a stemplot: 1)Truncate (or “trim”) the data to appropriate level of accuracy. 2)Separate each observation into a stem, consisting of all but the final (rightmost) digit, and a leaf, which is the remaining final digit. Stems may have as many digits as needed, but each leaf contains a single digit. 3)Write the stems in a column with the smallest value at the top, and draw a vertical line at the right of this column. 4)Write each leaf in the row to the right of its stem, in increasing order out from the stem. Leaves should be aligned in vertical columns. (Why?) STEMLEAVES

Percent of Hispanic residents in each of the 50 states Step 2: Assign the values to stems and leaves Step 1: Sort the data

Stemplots are quick and dirty histograms that can easily be done by hand, therefore very convenient for back of the envelope calculations. However, they are rarely found in scientific publications. Stemplots versus histograms

1)Advantages of Stemplots  Quick to do by hand (no frequency table needed)  Maintains all the numerical data  Good for comparing two distributions (using back-to-back plots) 2)Disadvantages  Not as crisp visually (compared to a graphics-quality histogram)  Most people like numerical scale on horizontal axis. Stemplots versus histograms

1)Stem-splitting  Use to increase the number of stems  Create one stem for leaves 0-4  Second stem for leaves 5-9 2)Back-to-back Stemplots  Use to compare two data sets measured on same scale (e.g., male vs. female heights).  Use single stem with leaves on opposite sides for different data sets. Variations on stemplots

Ways to chart quantitative data  Line graphs: time plots Use when there is a meaningful sequence, like time. The line connecting the points helps emphasize any change over time.  Histograms and stemplots These are summary graphs for a single variable. They are very useful to understand the pattern of variability in the data.  Other graphs to reflect numerical summaries (see Chapter 1.2)

Line graphs: time plots A trend is a rise or fall that persist over time, despite small irregularities. In a time plot, time always goes on the horizontal, x axis. We describe time series by looking for an overall pattern and for striking deviations from that pattern. In a time series: A pattern that repeats itself at regular intervals of time is called seasonal variation.

Retail price of fresh oranges over time This time plot shows a regular pattern of yearly variations. These are seasonal variations in fresh orange pricing most likely due to similar seasonal variations in the production of fresh oranges. There is also an overall upward trend in pricing over time. It could simply be reflecting inflation trends or a more fundamental change in this industry. Time is on the horizontal, x axis. The variable of interest—here “retail price of fresh oranges”— goes on the vertical, y axis.

A time plot can be used to compare two or more data sets covering the same time period. The pattern over time for the number of flu diagnoses closely resembles that for the number of deaths from the flu, indicating that about 8% to 10% of the people diagnosed that year died shortly afterward from complications of the flu.

A picture is worth a thousand words, BUT There is nothing like hard numbers.  Look at the scales. Scales matter How you stretch the axes and choose your scales can give a different impression.

Why does it matter? What's wrong with these graphs? Careful reading reveals that: 1. The ranking graph covers an 11-year period, the tuition graph 35 years, yet they are shown comparatively on the cover and without a horizontal time scale. 2. Ranking and tuition have very different units, yet both graphs are placed on the same page without a vertical axis to show the units. 3. The impression of a recent sharp “drop” in the ranking graph actually shows that Cornell’s rank has IMPROVED from 15th to 6th... Cornell’s tuition over time Cornell’s ranking over time